Question: Simplify and expand the following expression: $ \dfrac{2z}{3z - 9}+\dfrac{4z}{z - 1} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3z - 9)(z - 1)$ Multiply the first term by $\dfrac{z - 1}{z - 1}$ $ \begin{align*} \dfrac{2z}{3z - 9} \times \dfrac{z - 1}{z - 1} & = \dfrac{(2z)(z - 1)}{(3z - 9)(z - 1)} \\ & = \dfrac{2z^2 - 2z}{(3z - 9)(z - 1)}\end{align*} $ Multiply the second term by $\dfrac{3z - 9}{3z - 9}$ $ \begin{align*} \dfrac{4z}{z - 1} \times \dfrac{3z - 9}{3z - 9} & = \dfrac{(4z)(3z - 9)}{(z - 1)(3z - 9)} \\ & = \dfrac{12z^2 - 36z}{(z - 1)(3z - 9)}\end{align*} $ Now we have: $ = \dfrac{2z^2 - 2z}{(3z - 9)(z - 1)} + \dfrac{12z^2 - 36z}{(z - 1)(3z - 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2z^2 - 2z + 12z^2 - 36z}{(3z - 9)(z - 1)} $ $ = \dfrac{14z^2 - 38z}{(3z - 9)(z - 1)}$ Expand the denominator: $ = \dfrac{14z^2 - 38z}{3z^2 - 12z + 9}$